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Kinematics

The Language ofMotion

What’s a Kinematic?

•Kinematics

is the science of describingthe motion of objects using words,diagrams, numbers, graphs, andequations. The goal of any study ofkinematics is to develop sophisticatedmental models which serve us indescribing (and ultimately, explaining) themotion of real-world objects.

In this lesson….

•In this lesson, we will investigate the words usedto describe the motion of objects. That is, we willfocus on thelanguage

of kinematics. The wordslisted in later slides are used with regularity todescribe the motion of objects. Your goal shouldbe to become very familiar with their meanings.You may click on any word now to investigate itsmeaning or proceed with the lesson in the orderlisted at the bottom of this page

Physics—Math or Science?

•Physics is a mathematical science-

that is, theunderlying concepts and principles have amathematical basis. Throughout the course ofour study of physics, we will encounter a varietyof concepts which have a mathematical basisassociated with them. While our emphasis willoften be upon the conceptual nature of physics,we will give considerable and persistentattention to its mathematical aspect.

Words and Quantities

•The motion of objects can be described bywords-

words such as distance, displacement,speed, velocity, and acceleration. Thesemathematical quantities which are used todescribe the motion of objects can be dividedinto two categories. The quantity is either avector or a scalar. These two categories can bedistinguished from one another by their distinctdefinitions:

Scalar and Vector REVISITED

•Scalars

are quantities which are fullydescribed by a magnitude alone.

•Vectors

are quantities which are fullydescribed by both a magnitude and adirection.

Check your understanding

•Check Your Understanding

•1. To test your understanding of this distinction, consider the following quantities listed below. Categorizeeach quantity as being either a vector or a scalar.

•Quantity

•a. 5 m

•

•b. 30 m/sec, East

•

•c. 5 mi., North

•

•d. 20 degrees Celsius

•

•e. 256 bytes

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•f. 4000 Calories

•

•

Distance v. Displacement

•Distance and Displacement

•Distance and displacement are two quantities which mayseem to mean the same thing, yet have distinctlydifferent definitions and meanings.

•Distance

is ascalar quantity

which refers to "how muchground an object has covered" during its motion.

•Displacement

is avector quantity

which refers to "howfar out of place an object is"; it is the object's change inposition.

•Now consider another example. The diagrambelow shows the position of a cross-countryskier at various times. At each of the indicatedtimes, the skier turns around and reverses thedirection of travel. In other words, the skiermoves from A to B to C to D.

•

Use the diagram to determine the resultingdisplacement and the distance traveled by theskier during these three minutes.

Answer

•The skier covers a distance of

•(180 m + 140 m + 100 m) =420 m

andhas a displacement of140 m, rightward.

AnotherExample

•Now for a final example. A football coach pacesback and forth along the sidelines. The diagrambelow shows several of coach's positions atvarious times. At each marked position, the coachmakes a "U-turn" and moves in the oppositedirection. In other words, the coach moves fromposition A to B to C to D.

•

What is the coach's resulting displacement anddistance of travel?

.

Speed v.Velocity

Yes, there really is adifference!

What is speed?

•Just as distance and displacement havedistinctly different meanings (despite theirsimilarities), so do speed and velocity.Speed

is ascalar quantity

which refers to"how fast an object is moving." A fast-moving object has a high speed while aslow-moving object has a low speed. Anobject with no movement at all has a zerospeed.

Velocity

•Velocity is a vector quantity. As such,velocity is "direction-aware." Whenevaluating the velocity of an object,one must keep track of direction. Itwould not be enough to say that anobject has a velocity of 55 mi/hr. Onemust include direction information inorder to fully describe the velocity ofthe object. For instance, you mustdescribe an object's velocity as being55 mi/hr,east. This is one of theessential differences between speedand velocity. Speed is a scalar anddoes notkeep track of direction;velocity is a vector and isdirection-aware.

Average Speed v. Instantaneous

•The instantaneous speed of an object is not to beconfused with the average speed. Average speed is ameasure of the distance traveled in a given period oftime; it is sometimes referred to as the distanceper

time ratio. Suppose that during your trip to school, youtraveled a distance of 5 miles and the trip lasted 0.2hours (12 minutes). The average speed of your carcould be determined as avg. speed = distance/time

Average Velocity

For you to try:

•While on vacation, Lisa Carr traveled atotal distance of 440 miles. Her trip took 8hours. What was her average speed?

Answer

•To compute her average speed, we simplydivide the distance of travel by the time of travel.

•That was easy! Lisa Carr averaged a speed of55 miles per hour. She may not have beentraveling at a constant speed of 55 mi/hr. Sheundoubtedly, was stopped at some instant intime (perhaps for a bathroom break or for lunch)and she probably was going 65 mi/hr at otherinstants in time. Yet, she averaged a speed of55 miles per hour.

of 12meters in 24 seconds; thus, her average speedwas 0.50 m/s. However, since her displacementis 0 meters, her average velocity is 0 m/s.Remember that thedisplacement

refers to thechange in position and the velocity is basedupon this position change. In this case of theteacher's motion, there is a position change of 0meters and thus an average velocity of 0 m/s.

•

Return of the Skier Example

•Here is another example similar to what was seen beforein the discussion ofdistance and displacement. Thediagram below shows the position of a cross-countryskier at various times. At each of the indicated times, theskier turns around and reverses the direction of travel. Inother words, the skier moves from A to B to C to D.

•

Use the diagram to determine the average speed andthe average velocity of the skier during these threeminutes.

Answer

•

•The skier has an average speed of

•(420 m) / (3 min) =140 m/min

and anaverage velocity of

•(140 m, right) / (3 min) =46.7 m/min, right

Back to the coach example

•And now for the last example. A football coach pacesback and forth along the sidelines. The diagram belowshows several of coach's positions at various times. Ateach marked position, the coach makes a "U-turn" andmoves in the opposite direction. In other words, thecoach moves from position A to B to C to D.

•

What is the coach's average speed and averagevelocity? When finished, click the button to view theanswer

Answer

•Seymour has an average speed of

•(95 yd) / (10 min) =9.5 yd/min

and anaverage velocity of

•(55 yd, left) / (10 min) =5.5 yd/min, left

Summary

•In conclusion, speed and velocity are kinematicquantities which have distinctly differentdefinitions. Speed, being ascalar quantity, is thedistance

(a scalar quantity) per time ratio. Speedisignorant of direction. On the other hand,velocity isdirection-aware. Velocity, thevectorquantity, is the rate at which the positionchanges. It is thedisplacement

or positionchange (a vector quantity) per time ratio.

Acceleration

Definition

•Acceleration

is avector quantity

which isdefined as "the rate at which an objectchanges itsvelocity." An object isaccelerating if it is changing its velocity.

What does acceleration mean?

•Sports announcers will occasionally say that a person isaccelerating if he/she is moving fast. Yet accelerationhas nothing to do with going fast. A person can bemoving very fast, and still not be accelerating.Acceleration has to do with changing how fast an objectis moving. If an object is not changing its velocity, thenthe object is not accelerating. The data at the right arerepresentative of a northward-moving accelerating object-

the velocity is changing with respect to time. In fact, thevelocity is changing by a constant amount-

10 m/s-

ineach second of time. Anytime an object's velocity ischanging, that object is said to be accelerating; it has anacceleration.

Observe the animation of the three cars below. Which car or cars(red, green, and/or blue) are undergoing an acceleration? Studyeach car individually in order to determine the answer. Ifnecessary, review the definition ofacceleration.

Answers

•The green and blue cars are acceleratingwhile the red car is going at the samespeed throughout the animation. Thegreen and blue cars speed up.

Constant Acceleration

•Sometimes an accelerating object will change its velocityby the same amount each second. As mentioned in theabove paragraph, the data above show an objectchanging its velocity by 10 m/s in each consecutivesecond. This is referred to as aconstant acceleration

since the velocity is changing by a constant amount eachsecond. An object with a constant acceleration shouldnot be confused with an object with a constant velocity.Don't be fooled! If an object is changing its velocity-whether by a constant amount or a varying amount-

then it is an accelerating object. And an object with aconstant velocity is not accelerating

of the acceleration. The process can be reversedby taking successivederivatives.

•On the left hand side above, the constant acceleration is integrated toobtain the velocity. For this indefinite integral, there is a constant ofintegration. But in this physical case, the constant of integration has a verydefinite meaning and can be determined as an intial condition on themovement. Note that if you set t=0, then v = v0, the initial value of thevelocity. Likewise the further integration of the velocity to get an expressionfor the position gives a constant of integration. Checking the case where t=0shows us that the constant of integration is the initial position x0. It is true asa general property that when you integrate a second derivative of a quantityto get an expression for the quantity, you will have to provide the values oftwo constants of integration. In this case their specific meanings are theinitial conditions on the distance and velocity.

Free Falling Objects

•A falling object for instance usually accelerates as it falls. If we wereto observe the motion of afree-falling object

(free fall motion

will bediscussed in detail later), we would observe that the object averagesa velocity of 5 m/s in the first second, 15 m/s in the second second,25 m/s in the third second, 35 m/s in the fourth second, etc. Our free-falling object would be constantly accelerating. Given these averagevelocity values during each consecutive 1-second time interval, wecould say that the object would fall 5 meters in the first second, 15meters in the second second (for a total distance of 20 meters), 25meters in the third second (for a total distance of 45 meters), 35meters in the fourth second (for a total distance of 80 meters afterfour seconds). These numbers are summarized in the table below.

Time Interval

Ave. Velocity DuringTime Interval

Distance TraveledDuring Time Interval

Total Distance Traveledfrom 0 s to End of TimeInterval

0-

1 s

5 m/s

5 m

5 m

1-2 s

15 m/s

15 m

20 m

2-

3

25 m/s

25 m

45 m

3-

4 s

35 m/s

35 m

80 m

More on Free Fall

•This discussion illustrates that afree-falling object

whichis accelerating at a constant rate will cover differentdistances in each consecutive second. Further analysisof the first and last columns of the data above reveal thatthere is a square relationship between the total distancetraveled and the time of travel for an object starting fromrest and moving with a constant acceleration. The totaldistance traveled is directly proportional to the square ofthe time. As such, if an object travels for twice the time, itwill cover four times (2^2) the distance; the total distancetraveled after two seconds is four times the total distancetraveled after one second.

Free fall (cont.)

•If an object travels for three times the time, thenit will cover nine times (3^2) the distance; thedistance traveled after three seconds is ninetimes the distance traveled after one second.Finally, if an object travels for four times thetime, then it will cover 16 times (4^2) thedistance; the distance traveled after fourseconds is 16 times the distance traveled afterone second. For objects with a constantacceleration, the distance of travel is directlyproportional to the square of the time of travel.

Calculating Aavg

•Calculating the Average Acceleration

•The average acceleration of any objectover a given interval of time can becalculated using the equation

Velocity-Time Data Table

•This equation can be used to calculate theacceleration of the object whose motion isdepicted by thevelocity-time data table

•Since acceleration is avector quantity, it willalways have a direction associated with it. Thedirection of the acceleration vector depends ontwo things:

•whether the object is speeding up or slowingdown

•whether the object is moving in the + or-

direction

•

General Rule of Thumb

•The general RULE OF THUMB is:

•If an object is slowing down, then itsacceleration is in the opposite direction ofits motion.

•Consider the two data tables below. Ineach case, the acceleration of the object isin the "+" direction. In Example A, theobject is moving in the positive direction(i.e., has a positive velocity) and isspeeding up. When an object is speedingup, the acceleration is in the samedirection as the velocity. Thus, this objecthas apositive acceleration.

•In each case, the acceleration of theobject is in the "-" direction. In Example C,the object is moving in the positivedirection (i.e., has a positive velocity) andis slowing down. According to our RULEOF THUMB, when an object is slowingdown, the acceleration is in the appositedirection as the velocity. Thus, this objecthas anegative acceleration.

In Example B, the object is moving in thenegative direction (i.e., has a negativevelocity) and is slowing down. According toour RULE OF THUMB, when an object isslowing down, the acceleration is in theopposite direction as the velocity. Thus,this object also has apositiveacceleration.

•In Example D, the object is moving in thenegative direction (i.e., has a negativevelocity) and is speeding up. When anobject is speeding up, the acceleration isin the same direction as the velocity. Thus,this object also has anegativeacceleration.

Check your understanding

•To test your understanding of the conceptof acceleration, consider the followingproblems and the corresponding solutions.Use the equation for acceleration todetermine the acceleration for thefollowing two motions.